I am wondering is it possible to have a M16 thread on a 15mm rod? How is this manufactured? Any links to more info or answer would be very helpful.
The minor diameter for an M16 Coarse Male thread is 13.55mm, therefore, yes - a rod of outer diameter 15mm could be threaded such that it would accept an M16 nut. The strength of this interface would be significantly reduced over a standard M16 thread/nut combination, however. Whether this matters depends on your application.
If you wish to add a threaded section to a smooth 15mm rod e.g. at one end, it may make more sense to turn down the diameter to 14mm at that end and thread it at M14. The overall diameter may go down, but the strength of the threaded connection will be higher.
As for "how this is manufactured" - We need more information to provide a helpful answer.
For small scale, you could simply run a die down an 15mm smooth rod - if you use a lathe (powered down) to keep everything concentric you will get an acceptable thread.
Or you could turn down the outer diameter of an M16 rod to 15mm
According to CJ Winter, the act of rolling threads on a rod requires a blank with a diameter 0.051 mm smaller than the major thread diameter. That's not much of a difference from a 16 mm diameter rod (15.949 mm) and certainly not close to the 15 mm you've queried.